Group #29596167 — subgroups

Subgroups — #29596167

Order 15616.

358 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840179	C₂	1	1	yes	maximal, characteristic
2	#21447591	C₂	1	2	yes	maximal, complemented
3	#21445869	C₂	1	1	yes	maximal, characteristic
4	#21443914	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1812	C₂	1	1	yes	maximal, characteristic
6	#16840179	—	2	4	no	—
7	C₂×#14519717	C₄	1	2	yes	complemented
8	C₂×#14519717	C₂²	1	1	yes	characteristic
9	C₂²×#11758185	C₄	1	2	yes	complemented
10	C₂²×#11758185	C₂²	1	1	yes	characteristic
11	C₂×#14519713	C₂²	1	2	yes	—
12	#16839746	—	2	1	no	—
13	#16839584	—	2	4	no	—
14	#16839520	—	2	1	no	—
15	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#427	C₂²	1	2	yes	—
17	C₆₁×#410	—	2	1	no	—
18	C₆₁×H(3,ℤ/4)	—	2	1	no	—
19	C₂×#11758194	—	2	2	no	—
20	#14519717	—	2	4	no	—
21	#14519717	D₄	1	4	yes	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	—	2	4	no	—
24	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	C₂×#11758185	C₂×C₄	1	2	yes	—
27	#14519713	—	2	2	no	—
28	#14519713	—	2	2	no	—
29	#14519696	—	4	1	no	—
30	#14519694	—	4	1	no	—
31	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
32	#131×C₆₁	—	2	2	no	—
33	#131×C₆₁	—	2	2	no	—
34	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
38	#117×C₆₁	—	2	2	no	—
39	#107×C₆₁	—	2	2	no	—
40	#106×C₆₁	—	2	2	no	—
41	#99×C₆₁	—	4	2	no	—
42	#41×C₆₁	—	2	2	no	—
43	#41×C₆₁	—	2	2	no	—
44	C₂×Q₈×C₆₁	SD₁₆	1	1	yes	characteristic
45	C₂×D₄×C₆₁	SD₁₆	1	2	yes	—
46	C₂×D₄×C₆₁	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₆₁	—	2	2	no	—
50	M₁₆×C₆₁	—	2	4	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₁	#41	1	2	yes	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₂²⋊C₄×C₆₁	—	4	2	no	—
60	C₄²×C₆₁	—	4	2	no	—
61	#11758194	—	4	2	no	—
62	#11758194	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	#11758185	—	2	1	no	—
65	#11758185	—	2	1	no	—
66	#11758185	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₆₁	#106	1	2	yes	—
68	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₆₁	—	2	2	no	—
70	D₄×C₆₁	—	2	4	no	—
71	D₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	8	2	no	—
75	C₂×C₄×C₆₁	#106	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#121	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#135	1	2	yes	—
80	C₈×C₆₁	—	2	2	no	—
81	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₆₁	#117	1	1	yes	characteristic
83	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
84	#92418	—	8	2	no	—
85	#12832	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	2	no	—
87	C₂²×C₆₁	—	4	1	no	—
88	C₂²×C₆₁	—	4	2	no	—
89	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
90	C₂²×C₆₁	#465	1	2	yes	—
91	C₄×C₆₁	—	2	2	no	—
92	C₄×C₆₁	—	16	2	no	—
93	C₄×C₆₁	#380	1	1	yes	characteristic
94	C₄×C₆₁	#427	1	1	yes	characteristic
95	C₄×C₆₁	#465	1	2	yes	—
96	C₂×#504	—	61	1	no	—
97	#2155	—	61	2	no	—
98	#1813	—	61	1	no	—
99	#1812	D₆₁	1	1	yes	characteristic
100	#1382	—	61	2	no	—
101	C₂×C₆₁	—	2	2	no	—
102	C₂×C₆₁	—	2	1	no	—
103	C₂×C₆₁	—	4	2	no	—
104	C₂×C₆₁	#1807	1	1	yes	characteristic
105	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
106	#504	—	122	4	no	—
107	C₂×M₃₂	—	61	1	no	—
108	C₂×M₃₂	—	61	2	no	—
109	C₂²×C₁₆	—	61	2	no	—
110	C₂²×C₁₆	—	61	1	no	—
111	C₂×#108	—	61	2	no	—
112	#427	C₂×D₆₁	1	2	yes	—
113	#413	—	122	1	no	—
114	#410	—	2	1	no	—
115	#365	—	122	4	no	—
116	#338	—	122	1	no	—
117	H(3,ℤ/4)	—	2	1	no	—
118	C₆₁	#12832	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	C₂²⋊D₆₁	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	122	2	no	—
123	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
124	C₂×M₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₆₁	1	2	yes	—
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	#117	—	2	2	no	—
128	M₃₂	—	61	4	no	—
129	M₃₂	—	122	4	no	—
130	C₂×C₁₆	—	61	2	no	—
131	C₂×C₁₆	—	61	1	no	—
132	C₂×C₁₆	—	61	1	no	—
133	C₂×C₁₆	—	122	4	no	—
134	C₂×C₁₆	—	122	2	no	—
135	#108	—	122	2	no	—
136	#108	—	122	2	no	—
137	#107	—	2	2	no	—
138	#106	—	2	2	no	—
139	#101	—	244	1	no	—
140	#100	—	244	1	no	—
141	#99	—	4	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#11758201	1	1	yes	characteristic
145	C₂×D₄	#11758200	1	2	yes	—
146	C₂×D₄	#11758201	1	1	yes	characteristic
147	C₂²×C₄	#11758196	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	244	2	no	—
152	M₁₆	—	244	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	244	2	no	—
156	C₂×C₈	#11758196	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₈	D₄×D₆₁	1	2	yes	—
160	C₂×C₈	#11758224	1	2	yes	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	4	2	no	—
164	C₁₆	—	61	4	no	—
165	C₁₆	—	122	1	no	—
166	C₁₆	—	122	1	no	—
167	C₂³	#14519708	1	2	yes	—
168	C₂³	C₂×#11758200	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	#14519708	1	2	yes	—
176	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
177	C₂×C₄	#14519749	1	2	yes	—
178	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
179	C₂×C₄	#14519797	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	488	2	no	—
182	C₈	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
183	C₈	#14519733	1	1	yes	characteristic
184	C₈	#14519754	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#14519708	1	1	yes	characteristic
189	C₂²	#16839989	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	16	2	no	—
192	C₄	#16839620	1	1	yes	characteristic
193	C₄	#16839798	1	1	yes	characteristic
194	C₄	#16839993	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#21445842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840179

C₂

yes

maximal, characteristic

#21447591

C₂

yes

maximal, complemented

#21445869

C₂

yes

maximal, characteristic

#21443914

C₂

yes

maximal, complemented

C₆₁×#1812

C₂

yes

maximal, characteristic

#16840179

—

C₂×#14519717

C₄

yes

complemented

C₂×#14519717

C₂²

yes

characteristic

C₂²×#11758185

C₄

yes

complemented

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839746

—

#16839584

—

#16839520

—

C₂×#131×C₆₁

C₂²

yes

characteristic

C₆₁×#427

C₂²

yes

—

C₆₁×#410

—

C₆₁×H(3,ℤ/4)

—

C₂×#11758194

—

#14519717

—

#14519717

D₄

yes

—

C₂×#11758185

—

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

—

—

—

—

C₂×#41×C₆₁

D₄

yes

characteristic

#131×C₆₁

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

D₄

yes

characteristic

C₂×M₁₆×C₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

#107×C₆₁

—

#106×C₆₁

—

#99×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

SD₁₆

yes

characteristic

C₂×D₄×C₆₁

SD₁₆

yes

—

C₂×D₄×C₆₁

SD₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂³×C₆₁

#106

yes

—

C₂³×C₆₁

C₂×SD₁₆

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#106

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

—

#92418

—

#12832

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#106

yes

characteristic

C₂²×C₆₁

#465

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

100

#1382

—

101

C₂×C₆₁

—

102

C₂×C₆₁

—

103

C₂×C₆₁

—

104

C₂×C₆₁

#1807

yes

characteristic

105

C₂×#131

C₂×D₆₁

yes

characteristic

106

#504

—

122

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂²×C₁₆

—

110

C₂²×C₁₆

—

111

C₂×#108

—

112

#427

C₂×D₆₁

yes

—

113

#413

—

122

—

114

#410

—

115

#365

—

122

—

116

#338

—

122

—

117

H(3,ℤ/4)

—

118

C₆₁

#12832

yes

characteristic, complemented, Sylow

119

C₂×#41

C₂²⋊D₆₁

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

122

—

123

C₂×M₁₆

Dic₁₂₂

yes

—

124

C₂×M₁₆

C₂²⋊D₆₁

yes

characteristic

125

C₂²×C₈

C₄×D₆₁

yes

—

126

C₂²×C₈

C₂²×D₆₁

yes

characteristic

127

#117

—

128

M₃₂

—

129

M₃₂

—

122

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

122

—

134

C₂×C₁₆

—

122

—

135

#108

—

122

—

136

#108

—

122

—

137

#107

—

138

#106

—

139

#101

—

244

—

140

#100

—

244

—

141

#99

—

142

#41

—

143

#41

—

144

C₂×Q₈

#11758201

yes

characteristic

145

C₂×D₄

#11758200

yes

—

146

C₂×D₄

#11758201

yes

characteristic

147

C₂²×C₄

#11758196

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

244

—

152

M₁₆

—

244

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

244

—

156

C₂×C₈

#11758196

yes

—

157

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

158

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

159

C₂×C₈

D₄×D₆₁

yes

—

160

C₂×C₈

#11758224

yes

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₁₆

—

165

C₁₆

—

122

—

166

C₁₆

—

122

—

167

C₂³

#14519708

yes

—

168

C₂³

C₂×#11758200

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#14519708

yes

—

176

C₂×C₄

C₂×#11758196

yes

characteristic

177

C₂×C₄

#14519749

yes

—

178

C₂×C₄

C₂×#11758201

yes

characteristic

179

C₂×C₄

#14519797

yes

—

180

C₈

—

181

C₈

—

488

—

182

C₈

C₂²⋊C₄×D₆₁

yes

characteristic

183

C₈

#14519733

yes

characteristic

184

C₈

#14519754

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#14519708

yes

characteristic

189

C₂²

#16839989

yes

—

190

C₄

—

191

C₄

—

192

C₄

#16839620

yes

characteristic

193

C₄

#16839798

yes

characteristic

194

C₄

#16839993

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#21445842

yes

characteristic

	subgroup	G/H
center	C₄	#16839798
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1812	C₂
socle	C₂×C₆₁	#1807
prime radical	C₂×#41×C₆₁	D₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1812	D₆₁
hypocenter	C₆₁	#12832
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#16839798

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1812

C₂

socle

C₂×C₆₁

#1807

prime radical

C₂×#41×C₆₁

D₄

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
2. #21447591	C₂	semidirect
4. #21443914	C₂	semidirect
7. C₂×#14519717	C₄	semidirect
9. C₂²×#11758185	C₄	semidirect
118. C₆₁	#12832	semidirect

H ≅ G/N

type

2. #21447591

C₂

semidirect

4. #21443914

C₂

semidirect

7. C₂×#14519717

C₄

semidirect

9. C₂²×#11758185

C₄

semidirect

118. C₆₁

#12832

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16840179	2
#21447591	2
#21445869	2
#21443914	2
C₆₁×#1812	2
#12832	61

subgroup

index

C₆₁×#1812

Subgroups — #29596167

Conjugacy classes of subgroups

Complements

Maximal subgroups